Laplace and Mellin transform for reconstructing the probability distribution by a limited amount of information

A method for reconstructing the Probability Density Function (PDF) of a random variable using the Laplace transform is first introduced for one-sided PDFs. This approach defines new complex quantities, referred as Shifted Characteristic Functions, which allow the PDF to be computed using a classical Fourier series expansion. The method is then extended to handle double-sided PDFs by redefining the double-sided Laplace transform. This new definition remains applicable even when the integral in the inverse Laplace transform is discretized along the imaginary axis. For comparison, a new definition of double-sided Complex Fractional Moments based on Mellin transform is also introduced, addressing the singularity at zero that arises during PDF reconstruction.